13 Rao's score test: Recent asymptotic results

Abstract

While introducing the score test in 1948, Rao conjectured that it is likely to be locally more powerful than other tests like the likelihood ratio and Wald's tests. Recent work, based on higher-order asymptotics, indicates the truth of Rao's conjecture. This chapter reviews the work done in recent years in this area, and also summarizes some recent results on Bartlett-type adjustment for Rao's statistic in a private communication. For the scalar parameter case, under a one-sided alternative, R is evidently locally most powerful. As a consequence, the comparison among LR, R and W is of more interest when (1) the parameter θ is multi-dimensional or (2) the parameter is one-dimensional but the alternative is two-sided. Among the early authors, Peers (1971), Hayakawa (1975) and Harris and Peers (1980) studied these three tests, under the set-up of multi-dimensional θ and contiguous (multi-sided) alternatives, and reported their noncomparability. In particular, it was observed that, up to the second order of comparison, the local power function of none of these tests uniformly dominates those of the others. In consideration of this, subsequent authors treated the problem adopting the following two approaches: (1) to compare slightly modified versions of the tests, the modification being done in a meaningful way; and (2) to compare the tests in their original forms but using some reasonable criterion other than point-by-point power.